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Look up parallax measurement. This is essentially what we do to measure the distance to the nearest stars. The difference is that, since we can't make measurements from the sun (to measure the angle you've marked alpha), we measure the direction of the star from two different sides of the Earth's orbit, for example, measuring in June and then again in December. The small difference in these two angles, when combined with the diameter of the Earth's orbit, allows us to measure the distance to the star.